2.8 When and how should I remove unseen vertices?
Suggestion.  Archive you current file so it becomes your old file.  Get a fresh start and create a  new file. You may soon forget about the old file.



Composed from pieces with no excessive or time consuming vertex editing.  Mesh sequence in image. Top Left Gray UV Sphere.  Quarter sphere highlighted, result of quick edit.  Yellow top mesh is composed of 4 quarter spheres, duplicated, rotated 180 degrees and translated along an axis [XYZ].  Bottom row yellow mesh show edges joined with 2

[Bridge edge loops] ... there are many video tutorials on this

Cyan mesh show 4 [Bridge edge loops] with some edge color with the goal of more readability.  You may examine and verify and optimize these steps.  You can change the proportions in edit mode by grabbing [left, right] or [top, bottom] vertices.





Simple Yellow Cube.  Cyan form begins as cube,  modified with many loop cuts and a proportional edit.  All edges showing.  Image above.  Bevel Modifier is a possibility.



Start with a cube. The cube is already a closed well formed object and you can enhance it.  Add bevels or loop cuts and use proportional edit.  All of these terms can be researched here at BSE or the typical other locations you seem to have previously explored.  Your fresh start will be easier and quicker than the old file.





Torus Option. Image above.  


Top Gray Mesh Torus with Bevel in four places on 2 global axis [XY].  Your axis may be different.  Bevel Control B. Note the dark vertex density in four places.
Top Yellow Mesh.  [Push Pull] along a global [X] axis.  Select to vertical rings and invert selection then Push Pull.
Top Cyan Mesh.  [Push Pull] along a global [Y] axis.  Select to horizontal rings and invert selection then Push Pull.
Bottom yellow left mesh.  Delete inner faces. Only edge remains.
Bottom Cyan Mesh. [Grid Fill] on top edge hole. You may find it useful to hide the other vertices for this operation.
Bottom yellow right side mesh.  Side View.




I am not 100% clear on what you are trying to create but that in not 100% necessary.
There are always 33 ways to model something in Blender, some better and easier than others.  Your starting point in the question above is not a great learning sequence .  The older file seems to be creating a difficult and unnecessary stumbling block.

You might use Boolean operation, but that is a poor choice.